Convergence of Proximal Splitting Algorithms in CAT(k) Spaces and Beyond

TL;DR

This paper establishes the local linear convergence of fixed point iterations based on proximal mappings in CAT(k) spaces, demonstrating the necessity of linear metric subregularity for such convergence.

Contribution

It develops a theory for fixed point mappings that do not satisfy standard nonexpansiveness assumptions in p-uniformly convex spaces.

Findings

Proximal-based fixed point iterations converge locally linearly in CAT(k) spaces.

Linear metric subregularity is necessary for linear convergence.

The theory extends fixed point analysis beyond traditional nonexpansive mappings.

Abstract

In the setting of CAT(k) spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky-Mann relaxations, nonlinear projected-gradients) converge locally linearly under the assumption of linear metric subregularity. Linear metric subregularity is in any case necessary for linearly convergent fixed point sequences, so the result is tight. To show this, we develop a theory of fixed point mappings that violate the usual assumptions of nonexpansiveness and firm nonexpansiveness in p-uniformly convex spaces.